\(y = {\rm{\pi }} - 3\arccos \left( x \right)\), the y-axis and the line \(y = 2\).(a) Write down a definite integral to represent the area of R. [4](b) Calculate the area of R. [2]"/>

Effective GDC Use #1

Wednesday 12 September 2018

Approximate answers on Paper 2 exams

HL Paper 2 exam-like question #4

The region R is enclosed by the graph of \(y = {\rm{\pi }} - 3\arccos \left( x \right)\), the y-axis and the line \(y = 2\).

(a)    Write down a definite integral to represent the area of R.                                          [4]

(b)    Calculate the area of R.                                                                                              [2]

Solution:

(a)   area \( = \int_0^{0.928...} {\left[ {2 - \left( {{\rm{\pi }} - 3\arccos x} \right)} \right]} \,dx\)

(b)   area \( \approx 1.89\) \({\rm{unit}}{{\rm{s}}^2}\)


A question very similar to the one above was question #4 on the May 2017 HL Paper 2 exam (time zone 1).  The comment for the question that appeared in the subsequent Maths HL Subject Report for that exam session stated the following:

Part (a) was attempted by most candidates taking various approaches.  Limits were often incorrect and showed that many did not fully understand how to approach this question.

Part (b) was well done by those who had the correct expression in part (a); again, too many tried to integrate by hand instead of using their GDC.

It is worrying that many students did not do well on such a relatively straightforward question.  Not doing well on part (a) reveals a lack of conceptual understanding; whereas, not doing well on part (b) – given part (a) was answered correctly – indicates a poor awareness on the appropriate and effective use of a graphical display calculator (GDC).

On a Paper 2 exam (and Paper 3 for HL), it is essential that a student continually ask themselves whether the use of a GDC could be beneficial.  This is important for any mathematical work they are doing (homework, quizzes, unit tests) for which the use of a GDC is allowed.  Students need to read questions carefully – and keep a special eye out for whether a question requires an exact answer or not.  If an exact answer is not required, then an approximate answer – accurate to three significant figures unless stated otherwise – is perfectly acceptable.

For this question, part (b) did not ask for an exact answer, and it was only worth two marks.  From the Subject Report comment, it is clear that a significant number of students attempted to integrate manually which would have involved finding \(\int {\left[ {2 - \left( {{\rm{\pi }} - 3\arccos x} \right)} \right]} \,dx\).  Although it is possible to find the anti-derivative of \(\arccos x\) by applying a combination of ‘integration by parts’ and u-substitution, this would require quite a bit of time and certainly a quantity of working that would represent far more than just two marks.

But, it’s not enough just to point out the poor judgement of students.  It is likely that some students do not make effective use of their GDC when there is an opportunity to do so because they have not had adequate practice with questions that make them think hard about whether a GDC will be helpful or not.  In the same Subject Report, in the Paper 2 section Recommendations and guidance for the teaching of future candidates, it states:

Candidates need to recognize when they need to use algebra and calculus to gain marks and when using their GDC is a more effective and errorless way.

Hence, all of us teachers of IB Maths HL and SL need to strive to provide our students with regular opportunities (on homework, quizzes & tests) to develop skills in recognizing when the use of a GDC is – or is not – appropriate, efficient and wise.



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